Tools for optimizing dynamic systems using direct collocation.

## Introduction

opty utilizes symbolic descriptions of differential algebraic equations expressed with SymPy to form the constraints needed to solve optimal control and parameter identification problems using the direct collocation method and non-linear programming. In general, if one can express the continuous first order differential algebraic equations of the system as symbolic expressions opty will automatically generate a function to efficiently evaluate the dynamical constraints and a function that evaluates the sparse Jacobian of the constraints, which have been optimized for speed and memory consumption. The translation of the dynamical system description to the NLP form, primarily the formation of the constraints and the Jacobian of the constraints, manually is a time consuming and error prone process. opty eliminates both of those issues.

### Features

• Both implicit and explicit forms of the first order ordinary differential equations and differential algebraic equations are supported, i.e. there is no need to solve for the derivatives of the dependent variables.

• Backward Euler or Midpoint integration methods.

• Supports both trajectory optimization and parameter identification.

• Easy specification of bounds on free variables.

• Easily specify additional “instance” constraints.

• Automatic parallel execution using openmp if installed.

• Built with support of sympy.physics.mechanics and PyDy in mind.

## Installation

The required dependencies are as follows:

• python 3.8-3.11

• sympy >= 1.6.0

• ipopt >= 3.11 (Linux & OSX), >= 3.13 (Windows)

• numpy >= 1.19.0

• scipy >= 1.5.0

• cython >= 0.29.19

• cyipopt >= 1.1.0

To run all of the examples the following additional dependencies are required:

• matplotlib >= 3.2.0

• openmp

• pandas

• pydy >= 0.5.0

• pytables

The easiest way to install opty is to first install Anaconda (or Miniconda) and use the conda package manager to install opty and any desired optional dependencies from the Conda Forge channel, e.g. opty:

$conda install --channel conda-forge opty and the optional dependencies: $ conda install --channel conda-forge matplotlib openmp pandas pydy pytables yeadon

If you want a custom installation of any of the dependencies, e.g. Ipopt, you must first install Ipopt along with it’s headers. For example, on Debian based systems you can use the package manager:

$sudo apt-get install coinor-libipopt1v5 coinor-libipopt-dev or prebuilt binaries can be downloaded from https://www.coin-or.org/download/binary/Ipopt/. For customized installation (usually desired for performance) follow the instructions on the IPOPT documentation to compile the library. If you install to a location other than /usr/local on Unix systems you will likely have to set the LD_LIBRARY_PATH so that you can link to IPOPT when installing cyipopt. Once Ipopt is installed and accessible, install conda then create an environment: $ conda create -n opty-custom -c conda-forge cython numpy pip scipy sympy
$source activate opty-custom (opty-custom)$ pip install cyipopt  # this will compile cyipopt against the available ipopt
(opty-custom)$pip install opty If you want to develop opty, create a conda environment with all of the dependencies installed: $ conda config --add channels conda-forge
$conda create -n opty-dev python sympy numpy scipy cython ipopt cyipopt matplotlib pytables pydy pandas pytest sphinx numpydoc$ source activate opty-dev

Next download the opty source files and install with:

(opty-dev)$cd /path/to/opty (opty-dev)$ python setup.py develop

## Usage

There are several examples available in the examples directory. For example, the optimal torque to swing up a pendulum with minimal energy can be run with:

\$ python examples/pendulum_swing_up.py

## Funding

The work was partially funded by the State of Ohio Third Frontier Commission through the Wright Center for Sensor Systems Engineering (WCSSE), by the USA National Science Foundation under Grant No. 1344954, and by National Center of Simulation in Rehabilitation Research 2014 Visiting Scholarship at Stanford University, and the CZI grant CZIF2021-006198 and grant DOI https://doi.org/10.37921/240361looxoj from the Chan Zuckerberg Initiative Foundation (funder DOI 10.13039/100014989).

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